Let the real numbers () of the form (8) and (9) be given. Using these numbers, we construct the functions and by (14) and (15) and determine from the fundamental equation (13).
Now, let us construct the function by (3) and the function by (4). From , and have a derivative in both variables and these derivatives belong to .
Lemma 6 The following relations hold:
Proof Differentiating to x and y, (13), respectively, we get
It follows from (14) and (15) that
and using the fundamental equation (13), we obtain
Multiplying (31) on the left by B and we get
and multiplying (32) on the right by B and we have
Adding (36) and (37) and using (34), we find
From (33), we get
Integrating by parts and from (35)
is obtained. Substituting (40) into (38) and dividing by , we have
Multiplying (13) on the left by in the form of (4) and adding to (41)
is obtained. Setting
we can rewrite (42) as follows:
According to Lemma 4, the homogeneous equation (43) has only the trivial solution, i.e.
Differentiating (3) and multiplying on the left by B, we have
On the other hand, multiplying (3) on the left by and then integrating by parts and using (35), we find
It follows from (45) and (46) that
Taking into account (4) and (44),
is obtained. For , from (3) we get (30). □
Lemma 7 For each function , the following relation holds:
Proof It follows from (3) and (21) that
Using the expression
the fundamental equation (13) is transformed into the following form:
From (48), we get
and for the kernel we have the identity
and using (48) it is transformed into the following form:
Similarly, in view of (50), we have
According to (52),
It follows from (49) and (51) that
From (18) and the Parseval equality we obtain
Taking into account (54), we have
whence, by (52) and (53),
is obtained, i.e., (47) is valid. □
Corollary 8 For any function and , the following relation holds:
Proof (1) Let . Consider the series
Using Lemma 6 and integrating by parts, we get
Applying the asymptotic formulas in Theorem 1, is found. Consequently the series (57) converges absolutely and uniformly on . According to (55) and (58), we have
Since is arbitrary, is obtained, i.e.
(2) Fix and assume . Then, by virtue of (59),
The system is minimal in and consequently by (3), the system is minimal in . Hence and we obtain (56). □
Proof It is easily found that
According to (56), we get
We shall prove that for any n, . Assume the contrary, i.e. there exists m such that . Then for , it follows from (60) that . On the other hand, since as
. This contradicts the condition , . Hence, for any n. From (60), we have
Thus, we get , for any n. Since
we find , and then is obtained. □
Theorem 11 For the sequences () to be the spectral data for a certain boundary value problem of the form (1), (2) with , it is necessary and sufficient that the relations (8) and (9) hold.
Proof Necessity of the problem is proved in . Let us prove the sufficiency. Let the real numbers () of the form (8) and (9) be given. It follows from Lemma 6, Lemma 9, and Lemma 10 that the numbers () are spectral data for the constructed boundary value problem . The theorem is proved. □
The algorithm of the construction of the function by the spectral data () follows from the proof of the theorem:
By the given numbers () the functions and are constructed, respectively, by (14) and (15).
The function is found from (13).
is calculated by (4).